Calculus I · Limits and Continuity · lesson

Choosing Parameters for Piecewise Continuity

Learn Choosing Parameters for Piecewise Continuity with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-styl

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Where this chapter fits

Chapter 5: Continuity

Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.

Reading lens: Do the limit, the function value, and the surrounding domain fit together at the point or across the interval? Keep that question in view while reading Choosing Parameters for Piecewise Continuity; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects How to Repair a Removable Discontinuity to Two-Parameter Continuity Problems. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Set the left-hand expression, right-hand expression, and function value equal at a joining point; solve for unknown parameters.

Choosing Parameters in Piecewise Functions

At a piecewise join $x=a$ , continuity requires

\lim_{x\to a^-}f(x) =f(a) =\lim_{x\to a^+}f(x).

For ordinary polynomial pieces, substitute $x=a$ into both formulas and set the results equal.

Guided walkthrough

One parameter

Find $k$ so that

f(x)= \left\{\matrix{ kx+1,\quad x<2,\quad x+3,\quad x\ge2 }\right.

is continuous at $2$ .

Show worked solution

Left side at the join:

2k+1.

Right side and function value:

2+3=5.

Set them equal:

2k+1=5.

Subtract $1$ :

2k=4.

Divide by $2$ :

\boxed{k=2}.

Worked example

The parameter appears in both pieces

Find $a$ so that

g(x)= \left\{\matrix{ ax+1,\quad x<2,\quad x^2-a,\quad x\ge2 }\right.

is continuous at $2$ .

Show worked solution

The left-hand limit is

2a+1.

The right-hand limit and function value are

2^2-a=4-a.

Set them equal:

2a+1=4-a.

Add $a$ to both sides:

3a+1=4.

Subtract $1$ :

3a=3.

Therefore,

\boxed{a=1}.

Worked example

Exam-level: no parameter works

Find $k$ so that

h(x)= \left\{\matrix{ kx+2,\quad x<1,\quad x^2+k,\quad x\ge1 }\right.

is continuous at $1$ .

Show worked solution

The left-hand expression approaches

k+2.

The right-hand expression and function value equal

1+k.

Continuity would require

k+2=k+1.

Subtracting $k$ gives

2=1,

which is impossible. Therefore,

\boxed{\text{no value of }k\text{ works}}.

A parameter problem is not guaranteed to have a parameter solution. The algebra is allowed to reject the premise, rude though that may seem to a worksheet.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.

Chapter 5: Continuity

Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.

Choosing Parameters in Piecewise Functions

At a piecewise join

x=a

, continuity requires

\lim_{x\to a^-}f(x) =f(a) =\lim_{x\to a^+}f(x).

For ordinary polynomial pieces, substitute

x=a

into both formulas and set the results equal.

Guided walkthrough

One parameter

Find $k$ so that

f(x)= \left\{\matrix{ kx+1,\quad x<2,\quad x+3,\quad x\ge2 }\right.

is continuous at $2$ .

Show worked solution

Left side at the join:

2k+1.

Right side and function value:

2+3=5.

Set them equal:

2k+1=5.

Subtract $1$ :

2k=4.

Divide by $2$ :

\boxed{k=2}.

Worked example

The parameter appears in both pieces

Find $a$ so that

g(x)= \left\{\matrix{ ax+1,\quad x<2,\quad x^2-a,\quad x\ge2 }\right.

is continuous at $2$ .

Show worked solution

The left-hand limit is

2a+1.

The right-hand limit and function value are

2^2-a=4-a.

Set them equal:

2a+1=4-a.

Add $a$ to both sides:

3a+1=4.

Subtract $1$ :

3a=3.

Therefore,

\boxed{a=1}.

Worked example

Exam-level: no parameter works

Find $k$ so that

h(x)= \left\{\matrix{ kx+2,\quad x<1,\quad x^2+k,\quad x\ge1 }\right.

is continuous at $1$ .

Show worked solution

The left-hand expression approaches

k+2.

The right-hand expression and function value equal

1+k.

Continuity would require

k+2=k+1.

Subtracting $k$ gives

2=1,

which is impossible. Therefore,

\boxed{\text{no value of }k\text{ works}}.

A parameter problem is not guaranteed to have a parameter solution. The algebra is allowed to reject the premise, rude though that may seem to a worksheet.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

Chapter 5: Continuity

Choosing Parameters in Piecewise Functions

Related resources

Original instruction with traceable references.

Chapter 5: Continuity

Choosing Parameters in Piecewise Functions

Related resources

Original instruction with traceable references.